The Turkish airport in Turkish Cyprus is not recognized by Greek Cypiriot government, so if you do fly from Turkey to Cyprus, it is not considered a valid port of entry.
Thus you will get a stamp on a piece of paper, which you need to leave the country back to Turkey.
When traveling between the Turkish and Greek side of the island, the checkpoint guards will inspect your passport. As Greek Cyprus is part of the EU, EU laws require all EU Memberstates to allow free travel for any EU citizen. Likewise, due to agreements between the US, Cyprus and the EU, US citizens are too allowed free-travel. Citizens from other countries run the risk of being denied entry.
[0] https://travel.state.gov/content/travel/en/passports/have-pa...
There may well be other combinations of countries that justify a person having multiple passports.
That passport is since expired (it only had 3 years on it, at time of stamping), and my new passport book will get me wherever I happen to go. The souvenir sits safely in my safe.
If the traveller happened to remove that piece of paper once the returned from Cuba, so be it.
Citizens of many countries would be in big trouble if they have a stamp from Israeli customs on their passports. Customs apparently issue a separate document not to put a stamp on the passport as a result. This enabled many tourists who are "not supposed" to go there to nonetheless go there.